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Marc Lachièze-Rey – Blois 2007

The cosmological constant

Marc Lachièze-Rey (APC, Paris)

ΛMarc Lachièze-Rey – Blois 2007

The cosmological constant is not a problem

But a possible solution to explain various astrophysical data:

-age of the universe

- Galaxy formation

- Cosmography observations : SN data and others

…

Other possible solutions exist to these problems

like modified gravity, dark energy,…


Outline

History of ΛΛΛΛElements of cosmology

Observational evidence

The genuine cosmological constant

Other explanations


Historical elements: outline

• Einstein 1916 : no cosmological constant

• Einstein 1917: first relativistic model: ΛΛΛΛ

• 1930 (Slipher, Hubble, Lemaître) : cosmic expansion

--> Einstein renounces to ΛΛΛΛ

• Lemaître « saves » the “big bang” with ΛΛΛΛ

• Cosmic distance Recalibration : ΛΛΛΛ useless ?

--> 1970’s CDM paradigm: ΛΛΛΛ = 0

• problems : age , galaxy formation --> ΛΛΛΛ-cdm• SN’s observations

• ΛΛΛΛ denial : dark energy ?

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Original GR (Einstein, 1916)

Gravitation is described by the geometry of space-time

metric g --> Riemann --> Ricci --> Einstein tensor G(g)

The material content of the universe determines the geometry :

Einstein Equation : G (g) = χ T

T = energy-momentum tensor of material content

= the source of gravitation

Direction / Département / Service ou Laboratoire

6Unité qui présente / Réf présentation

Relativistic Cosmology (Einstein 1917)

GR is an ideal tool for cosmology :

--> find a space-time solution to describe the whole universe,

As a function of the average material content

(matter, radiation, etc.)

Einstein wants a cosmic model:

• without spatial infinite : closed spatial sections

• without spatial limit

• static (expansion unknown in 1917)

No such solution to Einstein equation as written

--> Einstein modifies its equation



New Einstein equation (1917)

G(g) = χ T G(g) = χ T + Λ g

New term Λ = cosmological constant- (no other term possible from mathematical consistency)

- absolute constant

- non material

- repulsive effect

- [almost] no effet of Λ for « local » problems (Solar System, galaxies, bh, …):

Only at cosmological scales.

- allows the desired solution = Einstein cosmological model

Static : attraction by matter balanced by repulsion by Λspace = a three-sphère : closed, no boundary !



Cosmic expansion (1930)

• Observations by Vesto Slipher, Edwin Hubble (Hubble law 1929)+ Theoretical work by Georges Lemaître (“Hubble law” 1927)

--> cosmic expansion --> Einstein renounces to ΛΛΛΛ

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Lemaitre (1931) : primordial atom (later : big bang)

wrong calibrations --> Age of the Universe < Age of the Earth

age problem

(also arguments from

galaxy formation)

Lemaître « saves »

the big bang with ΛQuickTime™ et un

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• Controversy (E/L): Einstein equation with or without Λ ?(Einstein will get out of the cosmological debate)


Historical elements.2• 1960’s : Cosmic distance Recalibration

--> age problem resolved without Λ : Λ useless ?

--> paradigm ( 1970’s) : cdm big bang models

Λ = ΩΛ= 0, Ωmatter = 1

• but • age of the universe: only marginally consistent

• galaxy formation --> Λ-cdm

One needs Λ

• 1990: Supernova observations (a classical cosmological test)

--> the expansion accelerates, exactly as predicted by Λ

Confirms the need of Λ : CDM --> CDM- Λ

• concordance : other observations also confirm the need of Λ

• More recently : « GR with Λ unsatisfactory »--> Search another explanation for cosmic data

(must mimick Λ)


Parameterisation of cosmology

Cosmic Dynamics

Spatial curvature :

Cosmological constant ΛΛΛΛ

Content = source of gravitation

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Cosmic dynamics

Expansion law: scale factor R(t)

expansion

rate

deceleration parameter

Third derivative --> parameter w

Beyond w = w(z)…

Present Value = Hubble constant= H0 ≈ 70 km /sec /Mpc

< 0 acceleration

> 0 deceleration

-.55


Spatial

curvature

k = sign

RC = spatial curvature radius

Fundamental

relation :


[true] cosmological Constant

Λ Λ Λ Λ constant by definition (required by mathematical consistency)

Cosmic units : (≈ 0.7)

Constant fundamental scale

RΛ=(Λ)-1/2 ∼ (3 Gpc)


Source of gravitation : any substance

energy density, pressure, equ. of State p = f(ρ)

Its cosmological influence depends on ρ +3p

Cosmological units : density parameter

In cosmology, ρ and p vary with z.It is always possible to define

From the cosmological point of view,

any substance is defined by Ω and weff

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Source of gravitation : any substance

Non relativistic matter (dust) p ≈ 0 : w=0

Radiation p = ρρρρ / 3 : w=1/3

Accelerating (Exotic !) ρρρρ +3p ΩΩΩΩΛΛΛΛ ~ .7

Even without SN’s observations


Observational evidences

– age of the Universe

–SNIa

–HST

–CMB

–LSS

–BAO du SDSS

– X-ray clusters

–Shear cosmique

–Effet Sachs-Wolfe intégré



Age of

the universe

tU = time duration since the univers was « very small ».

Finite by definition in big bang models.

Function of the cosmic parameters

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Larger than ages of earth, oldest stars… : > 12 Gyrs


Galaxy Formation

Galaxies need a long time to form.

If Λ =0, no sufficient time(already remarked by Lemaître).


Supernovas (SNIa) = luminosity distance measurements

Perlmutter 2003,Physics Today

Hyp : SNIa are standardizable candles

Dlum =L

4π f

Riess et al. 1998, AJ 116,,

1009 (High-Z SN Search)

Perlmutter et al 1999, ApJ

517, 565 (SCP)


Supernova cosmology project (Knop, Perlmuter)

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SN observations

• SuperNova Legacy Survey (SNLS) [14]. P. Astier,et al , A&A, 447,

31, (2006)

• ”gold” data set of supernovae

Riess et al. http://fr.arxiv.org/abs/astro-ph/0611572

• The ESSENCE Supernova Survey:

In its first four years, 102 type Ia SNe, at z from 0.10 to 0.78.


Essence

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.


SN results

Do we have good data,

good interpretation of them ?

• Are SNs good standard candles ?

• May some substance interact with

the photons and modify our

perception of SN data?

But concordance


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CMB alone measurements are degenerate in ΩM & ΩΛ(requires h)

--> Use CMB (WMAP-3 (Spergel et al. 06))

+ something else something else

• Different combinations of HST, SN, LSS, BAO, Shear, LTSW are conDifferent combinations of HST, SN, LSS, BAO, Shear, LTSW are consistent : sistent :

Overconstrained model : Overconstrained model :

• CMB + HST : ΩΛ = 0.758 +- 0.06HST Key Project measurement of the Hubble constant

(Freedman et al. 2001, ApJ 553, 47): h100 = 0.72 +-

0.08

• CMB + SNLS : ΩΛ = 0.719 +- 0.03

ΩΛ ~ 1−ΩM ~ 0.7


Galaxy clusters


Galaxy clusters

S.W. Allen et al. 2002 (arXiv:astro-ph/0205007v1)

X-ray gas mass fraction for a sample of

luminous X clusters (Chandra Observatory)

as a function of z

(independent mass confirmation from gravitational lensing studies.)

(gravitational instability)

( X-ray gas mass fractions -- >> an approximately

constant value at a radius r2500, (ρ =2500 ρcritical)

Combining with measurements of Ωbaryon and H0 (Hubble Key Project)

--> Ωm = 0.30+0.04, ΩΛ� = 0.95 Marc Lachièze-Rey – Blois 2007

BAO =baryonic acoustic oscillations

= Acoustic Peak

Luminous Red Galaxy (LRG) sample

from SDSS :

46 748 galaxies with 0.16

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BA0’s


∼150 Mpc

Cosmic

ruler


BA0’s

150 Mpc


SDSS

Aso 2DF

Project : Ly alpha

150 Mpc


BAOs



Assuming spatially flat

Conley et al. arXiv:astro-ph/0602411v2


Cosmic shear

Waerbeke et al. 2005, A&A 429, 75

VIRMOS

Decarte survey

VIRMOS survey

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Integrated Sachs Wolfe Effect

Cabré et al., astro-ph/0603690Correlation CMB -- galaxy distribution


Integrated Sachs Wolfe Effect


Correlation functions


Is the “good” Einstein equation with or without Λ ?

• Simplicity (Λ = 0) or generallity (Λ ≠ 0)

• ”Natural” Character of Λ :natural passage from Newton theory to GR

deformation theory : deformation parameter

--> Λ ≠ 0 : free parameter

Without decisive argument : observations.

Theoretical Arguments


Orders of magnitude

ρΛ ~ 2 ρmatter

This number is large : « cc problem »

This number is not large : « coincidence problem »

Cosmology

10-122MP4 == 10-54MEW

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Planck scale :

Particle physics

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Alternatives to Lambda


Explanations for cosmological

data• I - backreaction : physics is OK ; but new view on cosmology

• II - “genuine” Λ : “natural” explanation

Any other hypothesis (denial approach) requires new physics.

• III - modified gravity :

GR --> ? (strings, branes, non minimal coupling …?)

• IV - dark (black, exotic…) energy Requires a lot of fine tunings

Categorizing Different Approaches to the Cosmological Constant Problem S.Nobbenhuis arXiv:gr-qc/0411093


Back reaction(Kolb, Buchert…) : Averaging effect of inhomogeneities

Einstein equation : G(g) = χ Tbut < G(g) > ≠ χ < T >

« The cosmological effect of the neglected term in an inhomogeneous

universe can account for SN’s observed properties »

(and mimick cosmic acceleration in an homogeneous universe)

Is that true ? Does it explain other cosmological data ?

--> M N Célérier talk


Modified [classical] gravity

• Describe gravity with more fields : Non minimal coupling :

scalar-tensor, Brans - Dicke…

• Modify Lagrangian (extended curvature gravity) : R --> f(R)

• Conformal approaches (involve Weyl tensor)

Low scale tests, equivalence principle

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Strings and branes inspired explanations (More dimensions)

• String theories imply many new fields

• Scenarios from Brane physics :

gravity explores more dimensions

- ex.: DGP (Dvali-Gabadadze-Porrati) brane world model :

gravity is trapped on a four dimensional brane world at short distances,

but propagates into a higher dimensional space at large distances.

- ex.: Cyclic (ekpyroptic) universe (Steinhardt et al). :

Dynamics in a multi-dimensional world

All suggestions of modified gravity add degrees of freedom to GR,

to fit the data

Simplest modification : the true lambda explains everything

with only one (constant) parameter ! !


Anthropic view (string « landscape »)

Many « worlds » exist.

Each possess a value of Λ

If « many worlds » take some meaning

If some ensemble approach is possible (?)

If the proba distribution of Λ and the Proba distribution of life coincide : anthropic explanation

e.g. : proba of an observed universe = number of observers in that universe (how to calculate that ?)


Dark EnergyThe « joker of physics (and cosmology) » ?

= « substance » to mimic Λ (assumed to be zero)

Violates (at least strong) energy condition


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Dark energy :

• Must have repulsive gravitational effect : ρ + 3 p < 0

--> EXOTIC

Must be unclustered (otherwise detected)

No such thing in present physics! --> Invent it for the purpose. Two approaches

• 1) Just assume a “fluid” with the desired EOS : p(z) = w(z) ρ(z)

observations constrain [some integrals involving] w(z)

--> myriads of solutions, completely fine tuned

some authors parametrize w(z)

Ex : quinessence, Chaplyin gas …

• 2) Design a Lagrangian L

arbitrary functions to represent kinetic term and potential

Calculate EOS --> express with w(z) --> like the previous cases

--> constrain the arbitrary functions

Many solutions work ; the simplest = quintessence


From Uzan http://www2.iap.fr/pnc/JDEM/UzanJDEM-CNES150304.pdf


Dictionnary (first approach )

• « Decaying lambda » : Λ = Λ[a(t)] (ex. Λ = a-m ) or Λ = Λ[H(t)]

deduce ρ and w from momentum - energy conservation

•Quiessence = dark energy with a time-independent equ of state : w: Ct

• Kinessence = dark energy with a time-dependent equ of state

Ex. generalized Chaplyin gas p = K ρ−α

•Assume that the dynamics of the field adjusts in some way to that of matter/radiation

Tracker fields,

Holographic dark energy : ρΛ / ρmatter = r = Ct

• Phantom DE = w < -1 (requires modif of GR, eg non minimal coupling)

quintom = quintessence + phantom

•Viscous pressure p = P-Π, Π =Π [ div u]

(may arise from quantum effects, like particle creation)

See review and comparison in Silva e Costa and Makler (arXiv:astro-ph/0702418v1)

Connections among three roads to cosmic acceleration:

decaying vacuum, bulk viscosity, and nonlinear fluids


Dictionnary (Lagrangian approach)

•Quintessence = slowly evolving real Scalar field, with minimal coupling,

• K-essence = kinetic-energy-driven quintessence:

Non standard kinetic term in Lagrangian

(kinetic K- essence : only kinetic term)

• Phantom DE = w < -1 (requires modif of GR, eg non minimal coupling)

• quintom = quintessence + phantom

Klein

Gordon

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All these approaches require

fine adjustement

of arbitrary parameters and functions

to reproduce the data


KinessenceGeneric term for dark energy fluid with a time dependent equ of state,

written w = w(z) = p(z)/ρ(z)

acceleration ==> w < -1/3

•

•

Constraints -->


[ Kinessence : exemples of parametrization ]

SN Ia + SDSS + WMAP ==> w0 = -1.1, ΩM=0.25

Idem ==> w0 = -0.97, ΩM=0.28


W = w0+wa (1-a)


Kinessence : parametrization

Constraints from SN’s

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Quiessence

Dynamical age of the universe as a constraint on the parametrization of dark energy

equation of state, V. B. Johri P. K. Rath (arXiv:astro-ph/0603786v4)

time-independent equation of state : w = Ct

Ex • Network of non-interacting cosmic strings (w = -1/3)

• Domain walls (w = -2/3)

• Some forms of quintessence

• true

Observational constraints ==>


Quintessence

Imagine a real Scalar field with standard Lagrangian (minimal coupling)

Which gives the desired behaviour

w ≥ -1

(A priori w = w(z) --> a particular case of kinessence)

a common choice (Ratra-Peebles) V ∝ Φ-α


Dynamical evolution of Quintessence

Klein Gordon equation with damping (friction term) by cosmic expansion):

The field rolls down towards the minimum of V, at a rate driven by the shape of V

- Initially the kinetic energy dominates : `kination’.- It decreases faster than the radiation and the scalar field freeze- By now the potential energy becomes dominant (it was adjusted for that).

Any of these models requires generic fine tuning :

• the level of the potential (and/or mass) must reproduce the present lambda

• the shape must give the correct evolution (and be compatible with all cosmology)

If not, [m the friction term dominates and the field does not oscillate.m >> mH ==> the field would start to oscillate much earlier in the historyof the universe and would have reached a minimum long ago and would not affect the present universe.]


Quintessence

The potential gives the cosmic dynamics: w(z)

Then, adjust to fit cosmic data (e.g. acceleration)

But few constraints, limited precision, involve only integrals of w(z)


--> Many solutions possible

--> many quintessence models

Observational constraints

From ADVANCED TOPICS IN COSMOLOGY:

A PEDAGOGICAL INTRODUCTION

T. Padmanabhan

arXiv:astro-ph/0602117

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Particular cases of quintessence : Scaling solutions

The energy density of the scalar field evolves by mimicking some the

background fluid (ordinary matter or radiation)

--> energy density.: ρ�/ρmatter = Constante

For a set of initial conditions, acts as a dynamical attractor

Ex. Chaplyin gas : P = -A / ρ

Ex. tracker field …


Tracker fields (peculiar cases of quintessence)

Adjust potential so that the field density « adjusts » to that of matterapproaches an « attractor » solution

�designed to solve

the coincidence problem

QuickTime™ et undécompresseur TIFF (non compressé)sont requis pour visionner cette image.


kinetic-energy-driven quintessence = K - essence

Non trivial kinetic term in Lagrangian :

Pressure = Lagrangian density L = v(φ) F[K (φ)]

(K (φ) = usual kinetic energy term)

For some class of functions, admits solutions that track the equation of

state of the dominant type of matter (radiation in the early universe) until

pressure-less matter becomes dominant.

Then the k-essence begins to evolve toward cosmological constant

behavior (w= -1).

(An exemple : tachyonic field)

(Solutions giving w= Ct ≠ -1 are unstable)

Kinetic k-essence and Quintessence, Roland de Putter & Eric V. Linder (arXiv:0705.0400v1)


General warning :

data and models are often confronted with some a

priori (e.g. flat space hypothesis)

Dynamical Dark Energy or Simply Cosmic Curvature?

Clarkson, Cortes and Bassett (arXiv:astro-ph/0702670v1)

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Problems of dark energy

• energy of a quantum state not defined

• very unnatural mass (or potential) scale

• a lot of fine tuning is required

• no predictive power : every kind of models can reproduce observations (Padmanaban)

• almost no constraints

• coincidence problem

• dark energy assumes Λ=0 ; Why ?


Vacuum energy

[gravitational] Energy is not a well defined concept in quantum

physics : All calculation give ∞∞∞∞Only energy differences are defined (after regularization)

Note : Casimir effet is due to an energy difference

[Jaffe, …arXiv:hep-th/0503158] ________________________

Some hope ?

–Supersymmetry : bosons and fermion contributions compensate --> exactly 0

but supersymmetry breaking ? (how ?, bad scale)

– some cut-off for the energy integral (to be justified) -->

� finite value, but unnatural scale required : “ cc problem “


« Cosmological constant problem »

Why this number is so large ?

Maybe because acceleration has nothing to do with particle

physics ?

(Same problem than for, e.g., the masses of particles)

(despite the appellation, not a problem for the true

Cosmological constant )


« Coincidence problem »

Why ΩΛ of the same order than Ωmatter today ?

Why ΩΛ / Ωmatter not large ?

No explanation

(tracker type models are constructed in ad hoc fashion

to answer this problem)

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The coïncidence problem= « second cosmological constant problem ”.

Why ΩΛ ≈ 2 ΩmatterOr why RΛ ≈ c H

-1.

Why now ?


Is there any need for dark energy ?

Yes if some measurement gives w ≠ -1




A true cosmological constant ?

• Not a substance (a fluid) but a geometrical term

GR theory

• No unnatural value : given by observations

• Natural geometrical term in GR Lagrangian --> curvature of the GR vacuum

[ Einstein wish : vacuum of GR = nothing (no space-time)

Not true in present GR ]

No Λ :vacuum = Minkowski space-time with no curvature, no expansion

With Λ : vacuum = de Sitter constant curvature (Λ) space-time, expanding

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Arguments for a true cosmological constant

• Any change of theory introduces a new parameter : SR --> GR

• Deformation theory : Newton --> SR --> GR : c, Λ

• « cc problem » suggests that acceleraation has nothing to do

with particle physics

(in addition with vacuum energy problems, fine tuning required)

Padnamaban : Gravity’s immunity from vacuum energy

(--> new formulation of gr )


Two scales for the world ?

RΛ = Λ-2

= curvature radius of the fundamental state of GR :

vacuum (de sitter) space-time

--> Two fundamental length scales in the world:

Lplanck ≈ 10-30 cm et RΛ ≈ 1028 cm

(rem : Minkowski : RΛ =∞)


To conclude : Cosmology requires an accelerating factor

The true Λ explains all cosmological data. No new physics needed.

One unique arbitrary parameter solves the “problems”

Value not unnatural

Justify Λ ?

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